For any closed \(K\subseteq \mathbb {R}^n\) , recently all K-positivity preserver have been characterized, i.e., all linear operators \(T:\mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}[x_1,\dots ,x_n]\) such that \(Tp\ge 0\) on K for all \(p\ge 0\) on K. An important extension of polynomials \(\mathbb {R}[x_1,\dots ,x_n]\) with real coefficients are polynomials \(\mathbb {R}^{m\times m}[x_1,\dots ,x_n]\) with matrix coefficients. Non-negativity on K for matrix polynomials with Hermitian coefficients \(\textrm{Herm}_m\) is then \(p(x)\succeq 0\) for all \(x\in K\) . In the current work, we investigate linear operators \(T:\textrm{Herm}_m[x_1,\dots ,x_n]\rightarrow \textrm{Herm}_m[x_1,\dots ,x_n]\) . We focus on matrix K-positivity preserver, i.e., \(Tp\succeq 0\) on K for all \(p\succeq 0\) on K. For \(K=\mathbb {R}^n\) and compact sets \(K\subseteq \mathbb {R}^n\) , we give characterizations of matrix K-positivity preservers. We discuss the difference between the real and the matrix coefficient case and where our proof fails for general sets \(K\subseteq \mathbb {R}^n\) with \(K\ne \mathbb {R}^n\) and K non-compact.
